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2013 | 1 | 28-41
Tytuł artykułu

On the inverse of the adjacency matrix of a graph

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary and sufficient condition for two–vertex deleted subgraphs of G and of the graph ⌈(G−1) associated with G−1 to remain NSSDs is that the submatrices belonging to them, derived from G and G−1, are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSD.
Wydawca
Czasopismo
Rocznik
Tom
1
Strony
28-41
Opis fizyczny
Daty
otrzymano
2013-08-25
zaakceptowano
2013-11-13
online
2013-11-29
Twórcy
  • Department of Mathematics, University of Malta
  • Department of Mathematics, University of Malta
Bibliografia
  • [1] M. Fiedler, A characterization of tridiagonal matrices. Linear Algebra Appl., 2 (1969), 191–197.
  • [2] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin, 1986.
  • [3] W.H. Haemers, Interlacing eigenvalues and graphs. Linear Algebra Appl. 227–228 (1995), 593–616.[WoS]
  • [4] I. Sciriha, M. Debono, M. Borg, P. Fowler, and B.T. Pickup, Interlacing–extremal graphs. Ars Math. Contemp. 6(2)(2013), 261–278.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_spma-2013-0006
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